Question

In Exercises \(31-34,\) graph the piecewise-defined functions. $$f(x)=\left\\{\begin{array}{ll}{1,} & {x<0} \\ {\sqrt{x},} & {x \geq 0}\end{array}\right.$$

Short answer

For \(x < 0\), the graph of the function is a horizontal line at \(y = 1\); and for \(x \geq 0\), the graph will be a curve representative of the function \(y = \sqrt{x}\). The combination of these two drawn on the same plot gives the graph of the given piecewise-defined function.

Step-by-step solution

Identify the Functions

The function \(f(x)\) is defined in two parts based on the value of \(x\). These parts are \(1\) for when \(x < 0\) and \(\sqrt{x}\) for when \(x \geq 0\).

Sketching the Graph for \(x < 0\)

For all \(x\) less than \(0\), \(f(x) = 1\). This will be a straight line parallel to the \(x\)-axis at \(y = 1\), extending to the left from \(x = 0\). Note that at \(x = 0\), the function's value is not defined from this part, as \(x\) should be strictly less than \(0\). The point at \(x = 0\) for this part of the function is an open circle.

Sketching the Graph for \(x \geq 0\)

For all \(x\) values greater or equal to \(0\), \(f(x) = \sqrt{x}\). This is the standard square root function which starts at \(x = 0\), \(y = 0\), and extends towards the right. The point at \(x = 0\) in this part is a closed circle because \(0\) is included in this part of the domain.

Combining the graphs

Finally, the two parts of the graph are combined together to create the graph of the full function. The graph for \(x < 0\) is a horizontal line at \(y = 1\), and at \(x = 0\), there is an open circle. Then for \(x \geq 0\), the graph is that of \(y = \sqrt{x}\) extending rightwards, and at \(x = 0\), there is a closed circle which is at \(y = 0\) on the graph.

Key concepts

Piecewise-defined Functions

A piecewise-defined function is like a mathematical chameleon, showing different faces depending on the value of the input, or 'x'. These functions are multiple functions defined on different, non-overlapping intervals of the input. In simpler terms, a piecewise-defined function is a combination of two or more functions, each with its domain where it 'rules'.

Let's get to know the inhabitants of our piecewise function zoo. Take the function in the given exercise. For values of 'x' less than zero, the function transforms into a constant companion, always equal to 1. This behavior is indicative of the constant function. Now, for 'x' greater than or equal to zero, the function becomes a growing entity, following the square root function rules. It's essential for students to understand that the domain for which each function 'lives' determines how the graph will look in each section.

Square Root Function

The square root function, characterized by \( f(x) = \sqrt{x} \), is a gentle curve that starts its journey from the origin, \( (0, 0) \) and moves rightward, ever expanding and never looking back. Every point on this curve is like a patient answer to a whispered question: 'What number, squared, gives me 'x'?'.

Understanding the square root graph is fundamental because it often acts as a building block, a component in a larger function structure. It's essential to know that the domain of the square root function consists of all non-negative numbers since square roots of negative numbers are not defined in the set of real numbers. This nuanced detail impacts how the function behaves and how we graph it - always starting at zero and creeping to the right.

Sketching Graphs in Calculus

Sketching graphs in calculus is not just about drawing pretty pictures; it's a visual language for understanding function behaviors. The goal is to visualize the behavior of a function across its domain. Key features such as intercepts, continuity, and intervals of increase or decrease come to life through a well-drawn graph.

When graphing, accuracy is vital, particularly in areas where the function changes its rules, like the corner where piecewise-defined functions meet. Graphing also helps identify limits and asymptotic behaviors that pure algebra can't quite capture. Hence, it is an indispensable tool in a student's calculus arsenal.

Domain of Functions

Talking about the domain of functions is akin to discussing the 'habitat' where the function can thrive and exist. The domain is the complete set of 'x' values for which the function is defined and will output a real number. Excluding values that lead to undefined expressions (like division by zero or the square root of a negative number in real functions) is part of determining a function's domain.

For piecewise-defined functions, which have different rules for different intervals, identifying the domain for each 'piece' is crucial. It's essentially mapping out the territory where each function 'piece' has authority. Understanding this concept ensures that students don't get lost in the 'function forest' and can navigate the realms where each expression holds sway.